Finite Group Databasebeta

← #189863

Subgroups — #189863

Order 512.

1298 conjugacy classes of subgroups.

The subgroup lattice has 566 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1#34185C211yesmaximal, characteristic
2#21738C211yesmaximal, characteristic
3#13960C211yesmaximal, characteristic
4#8715C214yesmaximal, complemented
5#8714C214yesmaximal, complemented
6#8023C211yesmaximal, characteristic, complemented
7#8023C211yesmaximal, characteristic, complemented
8C2×#1294C211yesmaximal, characteristic, complemented
9C2×#1294C211yesmaximal, characteristic, complemented
10C22×C4×D4C2211yescharacteristic
11C22×C22⋊C8C2211yescharacteristic
12C22×C22⋊C8C2211yescharacteristic
13#2353—22no—
14#2353—22no—
15#2352C414yescomplemented
16#2352C414yescomplemented
17#2313—21no—
18#2312—21no—
19#2308—21no—
20C2×#386—21no—
21#2283C2212yes—
22#2280C2212yes—
23C2×#380C2211yescharacteristic
24C2×#380C2211yescharacteristic
25C2×#380C2211yescharacteristic
26#2273—21no—
27C22×#95C2211yescharacteristic
28#1869—22no—
29#1869—22no—
30#1817C2212yes—
31#1779C2212yes—
32#1441—21no—
33#1439—21no—
34#1434C2212yes—
35C22⋊#95—21no—
36C22⋊#95C2212yes—
37#1294C2214yes—
38#1294C2214yes—
39#1294C2214yes—
40#1294C2214yes—
41C23×D4D411yescharacteristic
42C24×C4—21no—
43C24×C4C2311yescharacteristic
44C24×C4C2311yescharacteristic
45C23×C8—21no—
46C23×C8—21no—
47C2×C4×D4—22no—
48C2×C4×D4—22no—
49C2×C4×D4—22no—
50C2×C4×D4—22no—
51C2×C4×D4C2×C412yes—
52C2×C4×D4C2×C412yes—
53C2×C4×D4D412yes—
54C2×C4×D4D412yes—
55C2×#117C2×C414yescomplemented
56C22×C4⋊C4—21no—
57C22×C4⋊C4C2×C412yes—
58C22×C4⋊C4C2×C412yes—
59C22×C4⋊C4D411yescharacteristic
60C22×C22⋊C4—21no—
61C22×C22⋊C4—21no—
62C22×C22⋊C4—21no—
63C22×C22⋊C4—22no—
64C22×C22⋊C4D411yescharacteristic
65C22×C22⋊C4D411yescharacteristic
66C22×C42C2311yescharacteristic
67C2×C22⋊C8—22no—
68C2×C22⋊C8—22no—
69C2×C22⋊C8—22no—
70C2×C22⋊C8—22no—
71C2×C22⋊C8—21no—
72C2×C22⋊C8—21no—
73C2×C22⋊C8—24no—
74C2×C22⋊C8C2312yes—
75C2×C22⋊C8C2312yes—
76#388—24no—
77#388—24no—
78#387—24no—
79#387—24no—
80#386—24no—
81#386—24no—
82#386—44no—
83#383—22no—
84#382C2×C414yescomplemented
85#381—22no—
86#381—22no—
87#380—21no—
88#380—21no—
89#380—21no—
90#380—21no—
91#380—22no—
92#380—22no—
93#380C2×C412yes—
94#380C2×C412yes—
95#380C2×C414yescomplemented
96#380C2312yes—
97#380C2312yes—
98#379—22no—
99C4×C4⋊C4C2×C414yescomplemented
100C4×C22⋊C4—22no—
101C4×C22⋊C4—22no—
102C4×C22⋊C4—22no—
103C4×C22⋊C4—24no—
104C4×C22⋊C4—24no—
105#376—22no—
106C2×#95—21no—
107C2×#95—21no—
108C2×#95—22no—
109C2×#95—22no—
110C2×#95—22no—
111C2×#95—22no—
112C2×#95—21no—
113C2×#95—22no—
114C2×#95C2312yes—
115C2×#95C2312yes—
116#336—24no—
117C25C2×D411yescharacteristic
118C25C2×D411yescharacteristic
119C22×D4—22no—
120C22×D4—22no—
121C22×D4—22no—
122C22×D4—22no—
123C22×D4—22no—
124C22×D4—22no—
125C22×D4C22⋊C412yes—
126C22×D4C22⋊C412yes—
127C23×C4—21no—
128C23×C4—21no—
129C23×C4—21no—
130C23×C4—21no—
131C23×C4—22no—
132C23×C4—22no—
133C23×C4—22no—
134C23×C4—22no—
135C23×C4—44no—
136C23×C4—42no—
137C23×C4—42no—
138C23×C4—42no—
139C23×C4—82no—
140C23×C4—82no—
141C23×C4—82no—
142C23×C4C22×C412yes—
143C23×C4C22×C411yescharacteristic
144C23×C4C22×C411yescharacteristic
145C23×C4C2×D411yescharacteristic
146C23×C4C2×D411yescharacteristic
147C23×C4C2×D411yescharacteristic
148C23×C4C2×D412yes—
149C23×C4C2×D411yescharacteristic
150C23×C4C2×D411yescharacteristic
151C23×C4C2×D411yescharacteristic
152C23×C4#4112yes—
153C23×C4#4112yes—
154C23×C4#4112yes—
155C23×C4#4112yes—
156C23×C4C2411yescharacteristic
157C22×C8—22no—
158C22×C8—22no—
159C22×C8—42no—
160C22×C8—42no—
161C22×C8—82no—
162C22×C8—82no—
163C4×D4—24no—
164C4×D4—24no—
165C4×D4—24no—
166C4×D4—24no—
167C4×D4—24no—
168C4×D4—24no—
169C4×D4—24no—
170C4×D4—24no—
171#117—84no—
172C2×C4⋊C4—22no—
173C2×C4⋊C4—22no—
174C2×C4⋊C4—22no—
175C2×C4⋊C4—22no—
176C2×C4⋊C4—42no—
177C2×C4⋊C4—82no—
178C2×C4⋊C4—82no—
179C2×C4⋊C4—82no—
180C2×C4⋊C4C22⋊C412yes—
181C2×C4⋊C4C22⋊C412yes—
182C2×C4⋊C4C22×C414yes—
183C2×C4⋊C4C22×C414yes—
184C2×C22⋊C4—22no—
185C2×C22⋊C4—22no—
186C2×C22⋊C4—22no—
187C2×C22⋊C4—22no—
188C2×C22⋊C4—22no—
189C2×C22⋊C4—22no—
190C2×C22⋊C4—22no—
191C2×C22⋊C4—22no—
192C2×C22⋊C4—24no—
193C2×C22⋊C4—24no—
194C2×C22⋊C4—42no—
195C2×C22⋊C4—42no—
196C2×C22⋊C4—42no—
197C2×C22⋊C4—42no—
198C2×C22⋊C4—44no—
199C2×C22⋊C4—44no—
200C2×C22⋊C4—42no—
201C2×C22⋊C4—42no—
202C2×C22⋊C4—42no—
203C2×C22⋊C4—82no—
204C2×C22⋊C4C22⋊C412yes—
205C2×C22⋊C4C22⋊C412yes—
206C2×C22⋊C4C22⋊C412yes—
207C2×C22⋊C4C22⋊C412yes—
208C2×C42—21no—
209C2×C42—21no—
210C2×C42—21no—
211C2×C42—21no—
212C2×C42—24no—
213C2×C42—24no—
214C2×C42C22×C412yes—
215C2×C42C2×D412yes—
216C22⋊C8—22no—
217C22⋊C8—22no—
218C22⋊C8—22no—
219C22⋊C8—22no—
220C22⋊C8—42no—
221C22⋊C8—42no—
222C22⋊C8—84no—
223#95—22no—
224#95—22no—
225#95—24no—
226#95—24no—
227#95—24no—
228#95—24no—
229#95—22no—
230#95—22no—
231#95—22no—
232#95—22no—
233#95—22no—
234#95—22no—
235#95—44no—
236#95—44no—
237#95—42no—
238#95—82no—
239C24—21no—
240C24—21no—
241C24—21no—
242C24—21no—
243C24—21no—
244C24—21no—
245C24—42no—
246C24—42no—
247C24—42no—
248C24—42no—
249C24C2×C22⋊C412yes—
250C24C2×C22⋊C411yescharacteristic
251C24C2×C22⋊C411yescharacteristic
252C24#12312yes—
253C24#12312yes—
254C24#13512yes—
255C24#13512yes—
256C24AGL(1,ℤ/8)12yes—
257C24#13712yes—
258C24C22×D411yescharacteristic
259C2×D4—24no—
260C2×D4—24no—
261C2×D4—24no—
262C2×D4—24no—
263C2×D4—24no—
264C2×D4—24no—
265C2×D4—44no—
266C2×D4—44no—
267C2×D4—44no—
268C2×D4—44no—
269C22×C4—21no—
270C22×C4—21no—
271C22×C4—21no—
272C22×C4—21no—
273C22×C4—21no—
274C22×C4—21no—
275C22×C4—21no—
276C22×C4—21no—
277C22×C4—21no—
278C22×C4—21no—
279C22×C4—22no—
280C22×C4—22no—
281C22×C4—22no—
282C22×C4—22no—
283C22×C4—24no—
284C22×C4—24no—
285C22×C4—24no—
286C22×C4—42no—
287C22×C4—42no—
288C22×C4—42no—
289C22×C4—44no—
290C22×C4—42no—
291C22×C4—42no—
292C22×C4—42no—
293C22×C4—42no—
294C22×C4—42no—
295C22×C4—44no—
296C22×C4—42no—
297C22×C4—42no—
298C22×C4—42no—
299C22×C4—48no—
300C22×C4—42no—
301C22×C4—42no—
302C22×C4—42no—
303C22×C4—44no—
304C22×C4—44no—
305C22×C4—42no—
306C22×C4—42no—
307C22×C4—42no—
308C22×C4—42no—
309C22×C4—42no—
310C22×C4—42no—
311C22×C4—42no—
312C22×C4—82no—
313C22×C4—82no—
314C22×C4—82no—
315C22×C4—82no—
316C22×C4—82no—
317C22×C4—82no—
318C22×C4—82no—
319C22×C4—82no—
320C22×C4—82no—
321C22×C4—82no—
322C22×C4—82no—
323C22×C4—82no—
324C22×C4—88no—
325C22×C4—88no—
326C22×C4—84no—
327C22×C4—84no—
328C22×C4C2×C22⋊C412yes—
329C22×C4C2×C22⋊C412yes—
330C22×C4C2×C22⋊C412yes—
331C22×C4C2×C22⋊C411yescharacteristic
332C22×C4C2×C22⋊C411yescharacteristic
333C22×C4#11712yes—
334C22×C4#11712yes—
335C22×C4#11712yes—
336C22×C4#11712yes—
337C22×C4#12312yes—
338C22×C4#12312yes—
339C22×C4#12312yes—
340C22×C4#12312yes—
341C22×C4#12312yes—
342C22×C4#12312yes—
343C22×C4C23×C411yescharacteristic
344C22×C4C22×D411yescharacteristic
345C22×C4C2×#4111yescharacteristic
346C22×C4C2×#4111yescharacteristic
347C22×C4C2×#4111yescharacteristic
348C22×C4C2×#4111yescharacteristic
349C2×C8—44no—
350C2×C8—44no—
351C2×C8—42no—
352C2×C8—42no—
353C2×C8—84no—
354C2×C8—84no—
355C4⋊C4—24no—
356C4⋊C4—24no—
357C4⋊C4—44no—
358C4⋊C4—44no—
359C4⋊C4—82no—
360C4⋊C4—82no—
361C4⋊C4—84no—
362C22⋊C4—24no—
363C22⋊C4—24no—
364C22⋊C4—24no—
365C22⋊C4—24no—
366C22⋊C4—42no—
367C22⋊C4—42no—
368C22⋊C4—42no—
369C22⋊C4—42no—
370C22⋊C4—44no—
371C22⋊C4—44no—
372C22⋊C4—84no—
373C22⋊C4—84no—
374C22⋊C4—88no—
375C22⋊C4—84no—
376C42—22no—
377C42—22no—
378C42—22no—
379C42—22no—
380C42—84no—
381C42—84no—
382C23—22no—
383C23—22no—
384C23—22no—
385C23—22no—
386C23—22no—
387C23—22no—
388C23—24no—
389C23—22no—
390C23—22no—
391C23—24no—
392C23—22no—
393C23—42no—
394C23—42no—
395C23—42no—
396C23—44no—
397C23—44no—
398C23—42no—
399C23—44no—
400C23—44no—
401C23—44no—
402C23—44no—
403C23—41no—
404C23—41no—
405C23—41no—
406C23—41no—
407C23—81no—
408C23—81no—
409C23—82no—
410C23—82no—
411C23—81no—
412C23—81no—
413C23#38112yes—
414C23#38112yes—
415C23#41612yes—
416C23#41612yes—
417C23#41712yes—
418C23#41712yes—
419C23#48214yes—
420C23#48214yes—
421C23#48514yes—
422C23#48514yes—
423C23C22×C22⋊C411yescharacteristic
424C23C2×#12311yescharacteristic
425C23C2×#12311yescharacteristic
426C23C2×#13511yescharacteristic
427C23C2×#13511yescharacteristic
428C23C2×AGL(1,ℤ/8)11yescharacteristic
429C23C2×#13711yescharacteristic
430D4—48no—
431D4—48no—
432C2×C4—22no—
433C2×C4—22no—
434C2×C4—22no—
435C2×C4—22no—
436C2×C4—24no—
437C2×C4—22no—
438C2×C4—22no—
439C2×C4—24no—
440C2×C4—22no—
441C2×C4—22no—
442C2×C4—22no—
443C2×C4—42no—
444C2×C4—42no—
445C2×C4—42no—
446C2×C4—41no—
447C2×C4—41no—
448C2×C4—41no—
449C2×C4—41no—
450C2×C4—42no—
451C2×C4—42no—
452C2×C4—42no—
453C2×C4—42no—
454C2×C4—42no—
455C2×C4—42no—
456C2×C4—42no—
457C2×C4—44no—
458C2×C4—44no—
459C2×C4—41no—
460C2×C4—41no—
461C2×C4—41no—
462C2×C4—41no—
463C2×C4—44no—
464C2×C4—44no—
465C2×C4—44no—
466C2×C4—44no—
467C2×C4—44no—
468C2×C4—44no—
469C2×C4—44no—
470C2×C4—44no—
471C2×C4—44no—
472C2×C4—82no—
473C2×C4—82no—
474C2×C4—81no—
475C2×C4—81no—
476C2×C4—81no—
477C2×C4—81no—
478C2×C4—82no—
479C2×C4—82no—
480C2×C4—82no—
481C2×C4—816no—
482C2×C4—82no—
483C2×C4—82no—
484C2×C4—82no—
485C2×C4—82no—
486C2×C4—82no—
487C2×C4—82no—
488C2×C4—82no—
489C2×C4—82no—
490C2×C4—164no—
491C2×C4—164no—
492C2×C4#38112yes—
493C2×C4#38112yes—
494C2×C4C2×#11711yescharacteristic
495C2×C4C2×#11711yescharacteristic
496C2×C4C2×#12311yescharacteristic
497C2×C4C2×#12311yescharacteristic
498C8—82no—
499C8—82no—
500C22—22no—
501C22—22no—
502C22—24no—
503C22—24no—
504C22—21no—
505C22—21no—
506C22—22no—
507C22—22no—
508C22—22no—
509C22—24no—
510C22—22no—
511C22—24no—
512C22—21no—
513C22—21no—
514C22—22no—
515C22—22no—
516C22—44no—
517C22—44no—
518C22—44no—
519C22—44no—
520C22—44no—
521C22—42no—
522C22—44no—
523C22—42no—
524C22—42no—
525C22—82no—
526C22—82no—
527C22—82no—
528C22—82no—
529C22#182514yes—
530C22#182514yes—
531C22C2×#38111yescharacteristic
532C22#288312yes—
533C22C2×#41611yescharacteristic
534C22C2×#41711yescharacteristic
535C22#290112yes—
536C22C2×#48211yescharacteristic
537C22C2×#48211yescharacteristic
538C22C2×#48511yescharacteristic
539C22C2×#48511yescharacteristic
540C4—21no—
541C4—21no—
542C4—21no—
543C4—21no—
544C4—22no—
545C4—22no—
546C4—42no—
547C4—42no—
548C4—42no—
549C4—84no—
550C4—84no—
551C4—84no—
552C4—162no—
553C2—21no—
554C2—21no—
555C2—21no—
556C2—21no—
557C2—24no—
558C2—24no—
559C2—82no—
560C2#1761512yes—
561C2#1762212yes—
562C2#2068911yescharacteristic
563C2C2×#182511yescharacteristic
564C2#2070611yescharacteristic
subgroupG/H
center, socleC23C22×C22⋊C4
commutatorC22×C4C23×C4
Frattini, cosocleC23×C4C24
prime radicalC23×D4D4
LCSG ▷ C22×C4 ▷ C22 ▷ C1
UCSC1 ◁ C23 ◁ C22×C4×D4 ◁ G
LECSG ▷ C23×C4 ▷ C22 ▷ C1

Complements

Splittings G = N ⋊ H with N normal; H ≅ G/N.

NH ≅ G/Ntype
4. #8715C2semidirect
5. #8714C2semidirect
6. #8023C2semidirect
7. #8023C2semidirect
8. C2×#1294C2semidirect
9. C2×#1294C2semidirect
15. #2352C4semidirect
16. #2352C4semidirect
55. C2×#117C2×C4semidirect
84. #382C2×C4semidirect
95. #380C2×C4semidirect
99. C4×C4⋊C4C2×C4semidirect

Maximal subgroups

By isomorphism type — maximal subgroups of the same type appear once, whether or not they are normal.

subgroupindex
#341852
#217382
#139602
#87152
#87142
#80232
C2×#12942